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AN INVERTED JACOB'S LADDER appears, and a Brand Corporation scientist starts climbing it to advocate confining and concealing nuclear explosions underground.

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ULAM: I think it is a very good idea to try to write a mathematical dictionary. First we must settle how many words to think about. Would you say two hundred or less?

ROTA: Two hundred! No. Ten, maybe!

ULAM: No, no. At least a hundred. They will have to be very diverse. It will be a long project. Logical words like but and even have a different character from words which have a topological or kinematical meaning like mix, find, search. Then there is another class of words like involve, intuitive, imaginary. There are many categories. I think we should have

The Mathematical Dictionary

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ULAM: And physical. Physics is almost completely mathematized now.

ROTA: I have already, and perhaps, and pending. They are close.

ULAM: Already is difficult mathematically. What about starting with but?

ROTA: Logicians claim but is the same as and.

ULAM: No! Its meaning is entirely different. How would you describe but logically? Something that leads us to a conclusion but does not? A disappointment in probability? A whole essay could be written about it. Someday there will be a tremendous theory devoted to its ramifications. It could be a germ like the word continuous. The study of topology

is nothing else but the study of the word continuous.

ROTA: When I was at Princeton, Alonzo Church gave a two-hour lecture on the meaning of but and and. It is now written up in his great Introduction to Mathematical Logic.

ULAM: So you see! And what does he say? I never read it. I knew he was a logician but did not know he did things like that. Now let's discuss things intelligently, professor.

ROTA: O.K. Let us begin with the word but. Stan?

ULAM: I would say that the word but suggests to me the following (we'll be more precise later): an element of an algebra whose elements are uttered sentences. I can imagine it as a point in a universe of points interpreted as sentences-physical facts. I see that it won't be easy to avoid circular definitions; we must not use the word but in developing a theory of but, right? The word but means that an element does not belong to a given set of points that was defined before. But-I am just saying this on purpose now-but expresses that an element belongs to a set which is similar or slightly larger than the already given set. Of course, I did not really need to use the word but in my explanation. However-Oh! I just used the word however; you see how hard it is to avoid these words? By the way, this poses another interesting philosophical problem, the fact that we cannot explain a mathematical ...

ROTA: Let's not digress.

ULAM: I just want to see what is in my mind. I do not have a perfect definition right away. Do you agree that but is an element which does not belong to a set that was defined before?

ROTA: Yes. Now let me try my definition. We have two sets, A and B, and a new relation between A and B which we will

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ULAM: I don't mind teaching, but I don't like to do it regularly. When I have to do something at a fixed hour, even if it is a pleasant dinner or cocktail, I fret. I hate not feeling completely free. But of course, being completely free immediately brings on a feeling of restlessness, of not knowing what to do!

Each of us has taught several thousand hours. If you think that a normal working year in America has about 2000 hoursan 8-hour day for about 250 days-that is quite a bit of your waking time, isn't it? But maybe it is not entirely waking time. One does it in a trance, partly asleep sometimes!

I am told I teach calculus well. It is possible, for I believe one should concentrate on the essence. One should not teach everything at a uniform level either. One should stress some important as well as some unimportant details on purpose— in a sense to follow the way I think the memory works.

When you remember a proof you remember a sequence of pleasant, unpleasant points, zeros and ones. Here comes a difficulty you try to remember, and you make an effort. Then you come to something that goes automatically and it is Then again a special zero, zero, zero. trick that has to be remembered. It is like going through a labyrinth.

ROTA: I am amazed at your labyrinth!

ULAM: I learn best from conversations. I love them, and that is how I learned physics in Los Alamos.

Some people are different in this respect. They prefer to learn slowly and methodically. How about you?

ROTA: I learn best when I am forced to do it.

ULAM: Speaking of being forced to learn, in Poland it happened several times that I announced that I would speak on a certain subject at a meeting of the mathematical society before I had a proof. I felt absolutely confident that once I had agreed to

speak, I would get a proof. It could have been an embarrassment otherwise.

On the other hand, when I look at a paper of mine which has been published, I discard it after one glance, from fear that I will discover that it is wrong. There is also this tiny gnawing doubt about whether the result is new or not. Yet even in a field about which I know nothing, I can always tell whether a theorem or a point of view is good or not. This feeling comes somehow from the way the quanti

John von Neumann

ULAM: Hot! What is the temperature?
ROTA: 80 or so.

ULAM: Pas possible! It must be the hottest day in thirty years. Which reminds me, once flying back to Los Alamos on Carco on a hot summer day, I opened the little window and my handkerchief flew out of the plane. Behind us there was a second plane carrying Johnny and others. What do you think the probability is that my handkerchief could have gotten enmeshed in the propeller of the other plane?

ROTA: Von Neumann was older than you. ULAM: Six, seven years.

ROTA: An older man!

ULAM: Yes. You know how it is. In the beginning the percentage was twenty or so; later it went down to ten.

ROTA: So you considered him a senior, and yet you made fun of him?

ULAM: Oh always! Of Banach too. I was always impudent.

ROTA: He did not treat you as someone younger?

fiers are arranged, from the tone or music of the piece.

Do you remember what Galois wrote in his final letter before his fatal duel? He wrote that in their publications mathematicians really conceal the way they obtain their results because the process of discovery is different from what appears in print. It is important to repeat this again and again.

Gainesville February 1974

ULAM: No. I don't think he knew anybody more intimately and vice versa, despite our difference in age. For a man of his stature he was curiously insecure, but his understanding, intelligence, mathematical breadth, and appreciation of what mathematics is for, historically and in the future, was unsurpassed. His immense work stands at the crossroads of the de

velopment of exact sciences. The rationalization of the idea of infinity-the life blood of its history-with its mysterious power to encode succintly and generally the properties of numbers and the patterns of geometry, received some of its definite formulations from his work. His ideas also advanced immeasurably the attempts to formalize the new, strange world of physics in the philosophically strange work of quantum theory. Fundamental ideas of how to start and proceed with the formal modes of operations and the scope of computing machines owe an immense debt to his work, though they still today give hints that are only dimly perceived about the workings of the nervous system and of the human brain itself.

Other mathematicians strike me as virtuosi who play their own special instru

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